Question: Is ${634572}$ divisible by $3$ ?
Explanation: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {634572}= &&{6}\cdot100000+ \\&&{3}\cdot10000+ \\&&{4}\cdot1000+ \\&&{5}\cdot100+ \\&&{7}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {634572}= &&{6}(99999+1)+ \\&&{3}(9999+1)+ \\&&{4}(999+1)+ \\&&{5}(99+1)+ \\&&{7}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {634572}= &&\gray{6\cdot99999}+ \\&&\gray{3\cdot9999}+ \\&&\gray{4\cdot999}+ \\&&\gray{5\cdot99}+ \\&&\gray{7\cdot9}+ \\&& {6}+{3}+{4}+{5}+{7}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${634572}$ is divisible by $3$ if ${ 6}+{3}+{4}+{5}+{7}+{2}$ is divisible by $3$ Add the digits of ${634572}$ $ {6}+{3}+{4}+{5}+{7}+{2} = {27} $ If ${27}$ is divisible by $3$ , then ${634572}$ must also be divisible by $3$ ${27}$ is divisible by $3$, therefore ${634572}$ must also be divisible by $3$.